Measurement, Latent Factors and Theory Construction

Confirmatory Factor Analysis in PyMC

cfa
sem
measurment
library(lavaan)
library(dplyr)
library(reticulate)
library(marginaleffects)
library(modelsummary)
library(ggplot2)
library(tidyr)
library(egg)
library(lme4)
library(semPlot)
library(tinytable)
library(kableExtra)
library(reshape2)
reticulate::py_run_string("import pymc as pm")

options(rstudio.python.installationPath = "/Users/nathanielforde/mambaforge/envs")
options("modelsummary_factory_default" = "tinytable")
options(repr.plot.width=15, repr.plot.height=8)

knitr::knit_engines$set(python = reticulate::eng_python)
options(scipen=999)
set.seed(130)
Warning

This is a work in progress. We intend to elaborate the choices related to analysing and understanding the data elicted through psychometrics surveys. Psychometric phenomena are inherently complex an imprecisely measured. Special attention needs to the assumptions built into the methodologies used to analyse this species of data.

Measurment and Measurement Constructs

Science is easier when there is a recipe. When there is some formula to follow, a procedure to adopt or routine to ape, you can out-source the responsibility for theory construction and methodological justification. One egregious pattern in this vein masks implausible nonsense with much vaunted mathematical gloss of “statistical significance”. Seen from 1000 feet, this is not surprising. Lip-service is paid to the idea of scientific method and we absolve ourselves of the requirement for principled justification and substantive argument. Evidence is marshaled in service to argument. It’s an absurd pretense to claim that data speaks for itself in this argument.

Data is found, gathered or maybe even carefully curated. In all cases there is need for a defensive posture, an argument that the data is fit-for-purpose. No where is this more clear than in psychometrics where the data is often derived from a strategically constructed survey aimed at a particular target phenomena. Some intuited, but not yet measured, concept that arguably plays a role in human action, motivation or sentiment. The “fuzziness” of the subject matter in psychometrics has had a catalyzing effect on the methodological rigour sought in the science. Survey designs are agonised over for correct tone and rhythm of sentence structure. Analysis steps are justified and tested under a wealth of modelling routines. Model architectures are defined and refined to better express the hypothesised structures in the data-generating process.

We will examine a smattering of choices available in the analysis of psychometric survey data. We will first step through these analysis routines using lmer, lavaan before switching to PyMC to demonstrate how to fit Confirmatory Factor Analysis models and Structural Equation Models in a Bayesian fashion using a probabilistic programming language.

The Data

The data is borrowed from work here demonstrating SEM modelling with Lavaan. We’ll load it up and begind some exploratory work.

df = read.csv('sem_data.csv')
df$ls_sum <- df$ls_p1 + df$ls_p2 + df$ls_p3
df$ls_mean <- rowMeans(df[ c('ls_p1', 'ls_p2', 'ls_p3')])

head(df) |> kable() |> kable_styling() |>  kable_classic(full_width = F, html_font = "Cambria")
ID region gender age se_acad_p1 se_acad_p2 se_acad_p3 se_social_p1 se_social_p2 se_social_p3 sup_friends_p1 sup_friends_p2 sup_friends_p3 sup_parents_p1 sup_parents_p2 sup_parents_p3 ls_p1 ls_p2 ls_p3 ls_sum ls_mean
1 west female 13 4.857143 5.571429 4.500000 5.80 5.500000 5.40 6.5 6.5 7.0 7.0 7.0 6.0 5.333333 6.75 5.50 17.58333 5.861111
2 west male 14 4.571429 4.285714 4.666667 5.00 5.500000 4.80 4.5 4.5 5.5 5.0 6.0 4.5 4.333333 5.00 4.50 13.83333 4.611111
10 west female 14 4.142857 6.142857 5.333333 5.20 4.666667 6.00 4.0 4.5 3.5 7.0 7.0 6.5 6.333333 5.50 4.00 15.83333 5.277778
11 west female 14 5.000000 5.428571 4.833333 6.40 5.833333 6.40 7.0 7.0 7.0 7.0 7.0 7.0 4.333333 6.50 6.25 17.08333 5.694444
12 west female 14 5.166667 5.600000 4.800000 5.25 5.400000 5.25 7.0 7.0 7.0 6.5 6.5 7.0 5.666667 6.00 5.75 17.41667 5.805556
14 west male 14 4.857143 4.857143 4.166667 5.20 5.000000 4.20 5.5 6.5 7.0 6.5 6.5 6.5 5.000000 5.50 5.50 16.00000 5.333333

The hypothetical dependency structure in this life-satisfaction data set posits a moderations relationship between scores related to life-satisfaction, parental and family support and self-efficacy. It is not a trivial task to be able to design a survey that can elicit answers plausibly mapped to each of these “factors” or themes, never mind finding a model of their relationship that can inform us as to the relative of impact of each on life-satisfaction outcomes.

Candidate Structure We can pull out the high level summary statistics to better observe the level of information in our metrics. These metrics are thematically clustered because the source survey deliberately targetted each of the underlying themes.

#| code-fold: true
#| code-summary: "Show the code"

datasummary_skim(df)|> 
 style_tt(
   i = 15:17,
   j = 1:1,
   background = "#20AACC",
   color = "white",
   italic = TRUE) |> 
 style_tt(
   i = 18:19,
   j = 1:1,
   background = "#2888A0",
   color = "white",
   italic = TRUE) |> 
 style_tt(
   i = 2:14,
   j = 1:1,
   background = "#17C2AD",
   color = "white",
   italic = TRUE)
tinytable_ocdtt0ymi9t18mtrz4el
Unique Missing Pct. Mean SD Min Median Max Histogram
ID 283 0 187.9 106.3 1.0 201.0 367.0
age 5 0 14.7 0.8 13.0 15.0 17.0
se_acad_p1 32 0 5.2 0.8 3.1 5.1 7.0
se_acad_p2 36 0 5.3 0.7 3.1 5.4 7.0
se_acad_p3 29 0 5.2 0.8 2.8 5.2 7.0
se_social_p1 24 0 5.3 0.8 1.8 5.4 7.0
se_social_p2 27 0 5.5 0.7 2.7 5.5 7.0
se_social_p3 31 0 5.4 0.8 3.0 5.5 7.0
sup_friends_p1 13 0 5.8 1.1 1.0 6.0 7.0
sup_friends_p2 10 0 6.0 0.9 2.5 6.0 7.0
sup_friends_p3 13 0 6.0 1.1 1.0 6.0 7.0
sup_parents_p1 11 0 6.0 1.1 1.0 6.0 7.0
sup_parents_p2 11 0 5.9 1.1 2.0 6.0 7.0
sup_parents_p3 13 0 5.7 1.1 1.0 6.0 7.0
ls_p1 15 0 5.2 0.9 2.0 5.3 7.0
ls_p2 21 0 5.8 0.7 2.5 5.8 7.0
ls_p3 22 0 5.2 0.9 2.0 5.2 7.0
ls_sum 98 0 16.2 2.1 8.7 16.4 20.8
ls_mean 97 0 5.4 0.7 2.9 5.5 6.9
N %
region east 142 50.2
west 141 49.8
gender female 132 46.6
male 151 53.4

Note how we’ve distinguished among the metrics for the “outcome” metrics and the “driver” metrics. Such a distinction may seem trivial, but it is only possible because we bring substantive knowledge to bear on the problem in the design of our survey and the postulation of the theoretical construct. Our data is a multivariate outcome with a large range of possible interacting effects and the patterns of realisation for our life-satisfaction scores may be quite different than the hypothesised structure. It is this open question that we’re aiming to uncover in the analysis of psychometrics surveys.

Sample Covariances and Correlations

drivers = c('se_acad_p1', 'se_acad_p2', 'se_acad_p3', 'se_social_p1', 'se_social_p2', 'se_social_p3', 'sup_friends_p1','sup_friends_p2', 'sup_friends_p3', 'sup_parents_p1' , 'sup_parents_p2' , 'sup_parents_p3', 'ls_p1', 'ls_p2', 'ls_p3')



plot_heatmap <- function(df, title="Sample Covariances", subtitle="Observed Measures") {
  heat_df = df |> as.matrix() |> melt()
  colnames(heat_df) <- c("x", "y", "value")
  g <- heat_df |> mutate(value = round(value, 2)) |>  ggplot(aes(x = x, y = y, fill = value)) +
    geom_tile() + geom_text(aes(label = value), color = "black", size = 4) +
   scale_fill_gradient2(
      high = 'dodgerblue4',
      mid = 'white',
      low = 'firebrick2'
    ) + theme(axis.text.x = element_text(angle=45)) + ggtitle(title, subtitle)
  
  g
}

g1 = plot_heatmap(cov(df[,  drivers]))

g2 = plot_heatmap(cor(df[,  drivers]), "Sample Correlations")

plot <- ggarrange(g1,g2, ncol=1, nrow=2);

Note the relatively strong correlations between measures of parental support and the life-satisfaction outcome ls_p3. Similarly, how the social self-efficacy scores seem similarly correlated with the secondary life satisfaction indicator ls_p2. These observed correlations merit some further investigation. We can plot the pairs of scatter plots.

df <- df |> 
  mutate(id = row_number()) 

# Prepare data to be plotted on the x axis
x_vars <- pivot_longer(data = df,
             cols = se_acad_p1:ls_p3,
             names_to = "variable_x",
             values_to = "x")

# Prepare data to be plotted on the y axis  
y_vars <- pivot_longer(data = df,
                       cols = se_acad_p1:ls_p3,
                       names_to = "variable_y",
                       values_to = "y") 

# Join data for x and y axes and make plot
full_join(x_vars, y_vars, 
          by = c("id"),
          relationship = "many-to-many") |>
  ggplot() + 
  aes(x = x, y = y) +
  geom_point() +
  facet_grid(c("variable_x", "variable_y"))  + ggtitle("Pair Plot of Indicator Metrics", 
                                                       "Comparing Against Life Satisfaction Scores")

The scatter plots among the highly correlated variables in the heatmap do seem to exhibit some kind of linear relationship with aspects of the life-satisfaction scores. We now turn to modelling these relationships to tease out some kind of inferential summary of that relationship.

Fit Initial Regression Models

To model these relationships we make use of the aggregated sum and mean scores to model the relationships between these indicator metrics and life-satisfaction. We fit a variety of models each one escalating in the number of indicator metrics we incorporate into our model of the life-satisfaction outcome. This side-steps the multivariate nature of hypothesised constructs and crudely amalgamates the indicator metrics. This may be more or less justified depending on how similar in theme the three outcome questions ls_p1, ls_p2, ls_p3 are in nature.

formula_sum_1st = " ls_sum ~ se_acad_p1  + se_social_p1 +  sup_friends_p1  + sup_parents_p1"
formula_mean_1st = " ls_mean ~ se_acad_p1  + se_social_p1 +  sup_friends_p1  + sup_parents_p1"

formula_sum_12 = " ls_sum ~ se_acad_p1  + se_acad_p2 +  se_social_p1 + se_social_p2 + 
sup_friends_p1 + sup_friends_p2  + sup_parents_p1 + sup_parents_p2"
formula_mean_12 = " ls_mean ~ se_acad_p1  + se_acad_p2 +  se_social_p1 + se_social_p2 + 
sup_friends_p1 + sup_friends_p2  + sup_parents_p1 + sup_parents_p2"


formula_sum = " ls_sum ~ se_acad_p1 + se_acad_p2 + se_acad_p3 + se_social_p1 +  se_social_p2 + se_social_p3 +  sup_friends_p1 + sup_friends_p2 + sup_friends_p3 + sup_parents_p1 + sup_parents_p2 + sup_parents_p3"
formula_mean = " ls_mean ~ se_acad_p1 + se_acad_p2 + se_acad_p3 + se_social_p1 +  se_social_p2 + se_social_p3 +  sup_friends_p1 + sup_friends_p2 + sup_friends_p3 + sup_parents_p1 + sup_parents_p2 + sup_parents_p3"
mod_sum = lm(formula_sum, df)
mod_sum_1st = lm(formula_sum_1st, df)
mod_sum_12 = lm(formula_sum_12, df)
mod_mean = lm(formula_mean, df)
mod_mean_1st = lm(formula_mean_1st, df)
mod_mean_12 = lm(formula_mean_12, df)

min_max_norm <- function(x) {
    (x - min(x)) / (max(x) - min(x))
}

df_norm <- as.data.frame(lapply(df[c(5:19)], min_max_norm))

df_norm$ls_sum <- df$ls_sum
df_norm$ls_mean <- df$ls_mean

mod_sum_norm = lm(formula_sum, df_norm)
mod_mean_norm = lm(formula_mean, df_norm)

models = list(
    "Outcome: sum_score" = list("model_sum_1st_factors" = mod_sum_1st,
     "model_sum_1st_2nd_factors" = mod_sum_12,
     "model_sum_score" = mod_sum,
     "model_sum_score_norm" = mod_sum_norm
     ),
    "Outcome: mean_score" = list(
      "model_mean_1st_factors" = mod_mean_1st,
     "model_mean_1st_2nd_factors" = mod_mean_12,
     "model_mean_score"= mod_mean, 
     "model_mean_score_norm" = mod_mean_norm
    )
    )

The classical presentation of regression models reports the coefficient weights accorded to each of the input variables. We present these models to highlight that the manner in which we represent our theoretical constructs has ramifications for the interpretation of the data generating process. In particular, note how different degrees of significance are accorded to the different variables depending on which variables are included.

#| code-fold: true
#| code-summary: "Show the code"

modelsummary(models, stars=TRUE, shape ="cbind") |> 
 style_tt(
   i = 2:25,
   j = 1:1,
   background = "#17C2AD",
   color = "white",
   italic = TRUE)
tinytable_pclqteqllnuzsr3iakja
Outcome: sum_score Outcome: mean_score
model_sum_1st_factors model_sum_1st_2nd_factors model_sum_score model_sum_score_norm model_mean_1st_factors model_mean_1st_2nd_factors model_mean_score model_mean_score_norm
+ p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001
(Intercept) 5.118*** 2.644** 2.094* 7.783*** 1.706*** 0.881** 0.698* 2.594***
(0.907) (0.985) (0.954) (0.658) (0.302) (0.328) (0.318) (0.219)
se_acad_p1 0.242 -0.034 -0.208 -0.800 0.081 -0.011 -0.069 -0.267
(0.147) (0.180) (0.192) (0.742) (0.049) (0.060) (0.064) (0.247)
se_social_p1 1.088*** 0.501* 0.355+ 1.846+ 0.363*** 0.167* 0.118+ 0.615+
(0.162) (0.204) (0.200) (1.039) (0.054) (0.068) (0.067) (0.346)
sup_friends_p1 0.125 -0.224+ -0.272+ -1.630+ 0.042 -0.075+ -0.091+ -0.543+
(0.088) (0.133) (0.150) (0.901) (0.029) (0.044) (0.050) (0.300)
sup_parents_p1 0.561*** 0.238+ 0.072 0.432 0.187*** 0.079+ 0.024 0.144
(0.100) (0.141) (0.143) (0.858) (0.033) (0.047) (0.048) (0.286)
se_acad_p2 0.448* 0.327 1.261 0.149* 0.109 0.420
(0.197) (0.202) (0.779) (0.066) (0.067) (0.260)
se_social_p2 0.756*** 0.509* 2.206* 0.252*** 0.170* 0.735*
(0.213) (0.219) (0.949) (0.071) (0.073) (0.316)
sup_friends_p2 0.369* 0.331* 1.490* 0.123* 0.110* 0.497*
(0.157) (0.160) (0.720) (0.052) (0.053) (0.240)
sup_parents_p2 0.370** 0.118 0.591 0.123** 0.039 0.197
(0.138) (0.144) (0.719) (0.046) (0.048) (0.240)
se_acad_p3 0.153 0.637 0.051 0.212
(0.174) (0.726) (0.058) (0.242)
se_social_p3 0.443** 1.771** 0.148** 0.590**
(0.161) (0.642) (0.054) (0.214)
sup_friends_p3 0.165 0.989 0.055 0.330
(0.130) (0.782) (0.043) (0.261)
sup_parents_p3 0.526*** 3.158*** 0.175*** 1.053***
(0.126) (0.754) (0.042) (0.251)
Num.Obs. 283 283 283 283 283 283 283 283
R2 0.399 0.467 0.517 0.517 0.399 0.467 0.517 0.517
R2 Adj. 0.390 0.451 0.496 0.496 0.390 0.451 0.496 0.496
AIC 1090.9 1064.7 1044.7 1044.7 469.1 442.9 422.9 422.9
BIC 1112.8 1101.2 1095.7 1095.7 491.0 479.4 473.9 473.9
Log.Lik. -539.455 -522.373 -508.341 -508.341 -228.548 -211.466 -197.434 -197.434
RMSE 1.63 1.53 1.46 1.46 0.54 0.51 0.49 0.49

We can similarly plot the coefficient values and their uncertainty highlighting how the representation or scaling of the variables impact the scale of the coefficient weights and therefore the surety of any subsequent claims.

models = list(
     "model_sum_1st_factors" = mod_sum_1st,
     "model_sum_1st_2nd_factors" = mod_sum_12,
     "model_sum_score" = mod_sum,
     "model_sum_score_norm" = mod_sum_norm,
     "model_mean_1st_factors" = mod_mean_1st,
     "model_mean_1st_2nd_factors" = mod_mean_12,
     "model_mean_score"= mod_mean,
     "model_mean_score_norm" = mod_mean_norm
    )

modelplot(models, coef_omit = 'Intercept') + geom_vline(xintercept = 0, linetype="dotted", 
                color = "black") + ggtitle("Comparing Model Parameter Estimates", "Across Covariates")

Significant Coefficients?

An alternative lens on these figures highlights the statistical significance of the coefficients. But again, these criteria are much abused. Significance at what level? Conditional on which representation?

g1 = modelplot(mod_sum, coef_omit = 'Intercept') +  aes(color = ifelse(p.value < 0.001, "Significant at 0.001", "Not significant at 0.001")) +
  scale_color_manual(values = c("grey", "blue")) + ggtitle("Significance of Coefficient Values", "At Different Levels")


g2 = modelplot(mod_mean, coef_omit = 'Intercept') +  aes(color = ifelse(p.value < 0.05, "Significant at 0.05", "Not significant at 0.05")) +
  scale_color_manual(values = c("grey", "black"))

plot <- ggarrange(g1,g2, ncol=2, nrow=1);

Aggregate Driver Scores

Perhaps we play with the feature representation and increase the proportion of significant indicators. Can we now tell a more definitive story about how parental support and social self-efficact are determinants of self-reported life-satisfaction scores? Let’s focus here on the sum score representation and add interaction effects.

df$se_acad_mean <- rowMeans(df[c('se_acad_p1', 'se_acad_p2', 'se_acad_p3')])
df$se_social_mean <- rowMeans(df[c('se_social_p1', 'se_social_p2', 'se_social_p3')])
df$sup_friends_mean <- rowMeans(df[c('sup_friends_p1', 'sup_friends_p2', 'sup_friends_p3')])
df$sup_parents_mean <- rowMeans(df[c('sup_parents_p1', 'sup_parents_p2', 'sup_parents_p3')])


formula_parcel_sum = "ls_sum ~ se_acad_mean + se_social_mean +
sup_friends_mean + sup_parents_mean " #sup_parents_mean*se_social_mean"

formula_parcel_sum_inter = "ls_sum ~ se_acad_mean + se_social_mean + 
sup_friends_mean + sup_parents_mean + sup_parents_mean*se_social_mean"

mod_sum_parcel = lm(formula_parcel_sum, df)
mod_sum_inter_parcel = lm(formula_parcel_sum_inter, df)

models_parcel = list("model_sum_score" = mod_sum_parcel,
     "model_sum_inter_score"= mod_sum_inter_parcel
     )

modelsummary(models_parcel, stars=TRUE)
tinytable_jxxnaq7a8a1a04s6zo8g
model_sum_score model_sum_inter_score
+ p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001
(Intercept) 2.728** -6.103+
(0.931) (3.356)
se_acad_mean 0.307+ 0.370*
(0.158) (0.158)
se_social_mean 1.269*** 2.859***
(0.175) (0.606)
sup_friends_mean 0.124 0.183+
(0.097) (0.099)
sup_parents_mean 0.726*** 2.242***
(0.099) (0.562)
se_social_mean × sup_parents_mean -0.292**
(0.107)
Num.Obs. 283 283
R2 0.489 0.503
R2 Adj. 0.482 0.494
AIC 1044.6 1039.0
BIC 1066.4 1064.5
Log.Lik. -516.288 -512.513
RMSE 1.50 1.48

What does definitive mean here? Is it so simple as more significant coefficents? Marginally better performance measures?

g1 = modelplot(mod_sum_parcel, coef_omit = 'Intercept') +  aes(color = ifelse(p.value < 0.05, "Significant at 0.05", "Not significant at 0.05")) +
  scale_color_manual(values = c("grey", "blue")) + ggtitle("Significance of Coefficient Values", "At Different Levels for Sum and Mean Scores Life Satisfaction ")


g2 = modelplot(mod_sum_inter_parcel, coef_omit = 'Intercept') +  aes(color = ifelse(p.value < 0.05, "Significant at 0.05", "Not significant at 0.05")) +
  scale_color_manual(values = c("grey", "blue"))

plot <- ggarrange(g1,g2, ncol=2, nrow=1);

This kind of brinkmanship is brittle. Any one of these kinds of choice can be justified but more often than not results from an suspect exploratory process. Steps down a garden of forking paths seeking some kind of story to justify an analysis or promote a conclusion. This post-hoc “seeking” is just bad science undermining the significance claims that accrue to reliable procedures. It warps the nature of testing procedure by corrupting the assumed consistency of repeatable trials. The guarantees of statistical significance attach to a conclusion just when the procedure is imagined replicable and repeated under identical conditions. By exploring the different representations and narrative criteria of adequacy we break those guarantees.

Exploratory and Confirmatory Modes

One of the things that psychometrics has pioneered well is the distinction between an exploratory and confirmatory models. This distinction, when made explicit, partially guards against the abuse of inferential integrity we see in more common work-flows. But additionally, models are often opaque - you may (as above) have improved some measure of model fit, changed the parameter weighting accorded to an observed feature, but that does that mean? Exploration of model architectures, design choices and feature creation is just how we come to understand the meaning of a model specification. Even in the simple case of regression we’ve seen how adding interaction terms changes the interpretability of a model. How then are we to stand behind uncertainty estimates accorded to parameter weights when we barely intuit the implications of a model design?

Marginal Effects

Evaluating inferences

pred <- predictions(mod_sum_parcel, newdata = datagrid(sup_parents_mean = 1:10 ))
pred1 <- predictions(mod_sum_inter_parcel, newdata = datagrid(sup_parents_mean = 1:10))

pred1 |> head() |> kableExtra::kable() |> kable_styling() |>  kable_classic(full_width = F, html_font = "Cambria")
rowid estimate std.error statistic p.value s.value conf.low conf.high se_acad_mean se_social_mean sup_friends_mean sup_parents_mean ls_sum
1 13.03023 0.5060063 25.75112 0 483.3544 12.03847 14.02198 5.237686 5.402768 5.929329 1 17.58333
2 13.69527 0.4074347 33.61341 0 820.4207 12.89671 14.49383 5.237686 5.402768 5.929329 2 17.58333
3 14.36031 0.3102480 46.28656 0 Inf 13.75224 14.96839 5.237686 5.402768 5.929329 3 17.58333
4 15.02536 0.2163209 69.45865 0 Inf 14.60137 15.44934 5.237686 5.402768 5.929329 4 17.58333
5 15.69040 0.1327619 118.18448 0 Inf 15.43019 15.95061 5.237686 5.402768 5.929329 5 17.58333
6 16.35544 0.0935245 174.87863 0 Inf 16.17214 16.53875 5.237686 5.402768 5.929329 6 17.58333
pred |> head() |> kableExtra::kable() |> kable_styling() |>  kable_classic(full_width = F, html_font = "Cambria")
rowid estimate std.error statistic p.value s.value conf.low conf.high se_acad_mean se_social_mean sup_friends_mean sup_parents_mean ls_sum
1 12.65390 0.4926138 25.68725 0 480.9813 11.68839 13.61940 5.237686 5.402768 5.929329 1 17.58333
2 13.37987 0.3953306 33.84477 0 831.6885 12.60504 14.15471 5.237686 5.402768 5.929329 2 17.58333
3 14.10585 0.2994218 47.11030 0 Inf 13.51899 14.69271 5.237686 5.402768 5.929329 3 17.58333
4 14.83183 0.2068081 71.71783 0 Inf 14.42649 15.23716 5.237686 5.402768 5.929329 4 17.58333
5 15.55780 0.1250396 124.42301 0 Inf 15.31273 15.80288 5.237686 5.402768 5.929329 5 17.58333
6 16.28378 0.0908259 179.28572 0 Inf 16.10576 16.46180 5.237686 5.402768 5.929329 6 17.58333

Regression Marginal Effects

g = plot_predictions(mod_sum_parcel, condition = c("se_social_mean"), type = "response") + ggtitle("Counterfactual Shift of Outcome: se_social_mean", "Holding all else Fixed")

g1 = plot_predictions(mod_sum_inter_parcel, condition = c("se_social_mean"), type = "response") + ggtitle("Counterfactual Shift of Outcome: se_social_mean", "Holding all else Fixed Interaction Model")

plot <- ggarrange(g,g1, ncol=1, nrow=2);

Hierarchical Models

formula_hierarchy_mean = "ls_mean ~ sup_parents_p1 + sup_parents_p2 + sup_parents_p3 + sup_friends_p1 + sup_friends_p2 + sup_friends_p3 + se_acad_p1 + se_acad_p2 + se_acad_p3 +
se_social_p1 + se_social_p2 + se_social_p3  + (1 | region)"

formula_hierarchy_sum = "ls_sum ~ sup_parents_p1 + sup_parents_p2 + sup_parents_p3 + sup_friends_p1 + sup_friends_p2 + sup_friends_p3 + se_acad_p1 + se_acad_p2 + se_acad_p3 +
se_social_p1 + se_social_p2 + se_social_p3 + (1 | region)"

hierarchical_mean_fit <- lmer(formula_hierarchy_mean, data = df, REML = TRUE)
boundary (singular) fit: see help('isSingular')
hierarchical_sum_fit <- lmer(formula_hierarchy_sum, data = df, REML = TRUE)
boundary (singular) fit: see help('isSingular')
g1 = modelplot(hierarchical_mean_fit, re.form=NA) +  aes(color = ifelse(p.value < 0.001, "Significant at 0.001", "Not significant at 0.001")) +
  scale_color_manual(values = c("grey", "blue")) + ggtitle("Significance of Coefficient Values", "At Different Levels")

g2 = modelplot(hierarchical_sum_fit, re.form=NA) +  aes(color = ifelse(p.value < 0.05, "Significant at 0.05", "Not significant 0.05")) +
  scale_color_manual(values = c("grey", "black"))


plot <- ggarrange(g1,g2, ncol=2, nrow=1);
Warning: Removed 2 rows containing missing values or values outside the scale range
(`geom_segment()`).
Removed 2 rows containing missing values or values outside the scale range
(`geom_segment()`).

modelsummary(list("hierarchical_mean_fit"= hierarchical_mean_fit,
                  "hierarchical_sum_fit"= hierarchical_sum_fit), 
             stars = TRUE) |> 
 style_tt(
   i = 2:25,
   j = 1:1,
   background = "#17C2AD",
   color = "white",
   italic = TRUE)
tinytable_6dqjw6ypw5guaj07q8f6
hierarchical_mean_fit hierarchical_sum_fit
+ p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001
(Intercept) 0.698* 2.094*
(0.318) (0.954)
sup_parents_p1 0.024 0.072
(0.048) (0.143)
sup_parents_p2 0.039 0.118
(0.048) (0.144)
sup_parents_p3 0.175*** 0.526***
(0.042) (0.126)
sup_friends_p1 -0.091+ -0.272+
(0.050) (0.150)
sup_friends_p2 0.110* 0.331*
(0.053) (0.160)
sup_friends_p3 0.055 0.165
(0.043) (0.130)
se_acad_p1 -0.069 -0.208
(0.064) (0.192)
se_acad_p2 0.109 0.327
(0.067) (0.202)
se_acad_p3 0.051 0.153
(0.058) (0.174)
se_social_p1 0.118+ 0.355+
(0.067) (0.200)
se_social_p2 0.170* 0.509*
(0.073) (0.219)
se_social_p3 0.148** 0.443**
(0.054) (0.161)
SD (Intercept region) 0.000 0.000
SD (Observations) 0.498 1.493
Num.Obs. 283 283
R2 Marg. 0.506 0.506
AIC 482.6 1075.9
BIC 537.3 1130.6
RMSE 0.49 1.46

Hierarchical Marginal Effects

g = plot_predictions(hierarchical_mean_fit, condition = c("sup_parents_p3", "region"), type = "response", re.form=NA) + ggtitle("Counterfactual Shift of Outcome: sup_parents_p3", "Holding all else Fixed")

g1 = plot_predictions(hierarchical_mean_fit, condition = c("sup_friends_p1", "region"), type = "response", re.form=NA) + ggtitle("Counterfactual Shift of Outcome: sup_friends_p1", "Holding all else Fixed")

g2 = plot_predictions(hierarchical_mean_fit, condition = c("se_acad_p1", "region"), 
                      type = "response", re.form=NA) + ggtitle("Counterfactual Shift of Outcome: se_acad_p1", "Holding all else Fixed")

plot <- ggarrange(g,g1,g2, ncol=1, nrow=3);

Confirmatory Factor Analysis

model_measurement <- "
# Measurement model
SUP_Parents =~ sup_parents_p1 + sup_parents_p2 + sup_parents_p3
SUP_Friends =~ sup_friends_p1 + sup_friends_p2 + sup_friends_p3
SE_Academic =~ se_acad_p1 + se_acad_p2 + se_acad_p3
SE_Social =~ se_social_p1 + se_social_p2 + se_social_p3
LS  =~ ls_p1 + ls_p2 + ls_p3
"

model_measurement1 <- "
# Measurement model
SUP_Parents =~ sup_parents_p1 + sup_parents_p2 + sup_parents_p3
SUP_Friends =~ a1*sup_friends_p1 + a2*sup_friends_p2 + a3*sup_friends_p3
SE_Academic =~ se_acad_p1 + se_acad_p2 + se_acad_p3
SE_Social =~ se_social_p1 + se_social_p2 + se_social_p3
LS  =~ ls_p1 + ls_p2 + ls_p3

a1 == a2 
a1 == a3
"


fit_mod <- cfa(model_measurement, data = df)
fit_mod_1<- cfa(model_measurement1, data = df)


cfa_models = list("full_measurement_model" = fit_mod, 
     "measurement_model_reduced" = fit_mod_1)
modelplot(cfa_models)
Warning: Removed 2 rows containing missing values or values outside the scale range
(`geom_segment()`).

summary(fit_mod, fit.measures = TRUE, standardized = TRUE) 
lavaan 0.6-18 ended normally after 49 iterations

  Estimator                                         ML
  Optimization method                           NLMINB
  Number of model parameters                        40

  Number of observations                           283

Model Test User Model:
                                                      
  Test statistic                               223.992
  Degrees of freedom                                80
  P-value (Chi-square)                           0.000

Model Test Baseline Model:

  Test statistic                              2696.489
  Degrees of freedom                               105
  P-value                                        0.000

User Model versus Baseline Model:

  Comparative Fit Index (CFI)                    0.944
  Tucker-Lewis Index (TLI)                       0.927

Loglikelihood and Information Criteria:

  Loglikelihood user model (H0)              -4285.972
  Loglikelihood unrestricted model (H1)      -4173.976
                                                      
  Akaike (AIC)                                8651.944
  Bayesian (BIC)                              8797.761
  Sample-size adjusted Bayesian (SABIC)       8670.921

Root Mean Square Error of Approximation:

  RMSEA                                          0.080
  90 Percent confidence interval - lower         0.067
  90 Percent confidence interval - upper         0.092
  P-value H_0: RMSEA <= 0.050                    0.000
  P-value H_0: RMSEA >= 0.080                    0.500

Standardized Root Mean Square Residual:

  SRMR                                           0.056

Parameter Estimates:

  Standard errors                             Standard
  Information                                 Expected
  Information saturated (h1) model          Structured

Latent Variables:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
  SUP_Parents =~                                                        
    sup_parents_p1    1.000                               0.935    0.873
    sup_parents_p2    1.036    0.056   18.613    0.000    0.969    0.887
    sup_parents_p3    0.996    0.059   16.754    0.000    0.932    0.816
  SUP_Friends =~                                                        
    sup_friends_p1    1.000                               1.021    0.906
    sup_friends_p2    0.792    0.043   18.463    0.000    0.809    0.857
    sup_friends_p3    0.891    0.050   17.741    0.000    0.910    0.831
  SE_Academic =~                                                        
    se_acad_p1        1.000                               0.695    0.878
    se_acad_p2        0.809    0.050   16.290    0.000    0.562    0.820
    se_acad_p3        0.955    0.058   16.500    0.000    0.664    0.829
  SE_Social =~                                                          
    se_social_p1      1.000                               0.638    0.843
    se_social_p2      0.967    0.056   17.248    0.000    0.617    0.885
    se_social_p3      0.928    0.067   13.880    0.000    0.592    0.741
  LS =~                                                                 
    ls_p1             1.000                               0.667    0.718
    ls_p2             0.778    0.074   10.498    0.000    0.519    0.712
    ls_p3             0.968    0.090   10.730    0.000    0.645    0.732

Covariances:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
  SUP_Parents ~~                                                        
    SUP_Friends       0.132    0.064    2.073    0.038    0.138    0.138
    SE_Academic       0.218    0.046    4.727    0.000    0.336    0.336
    SE_Social         0.282    0.045    6.224    0.000    0.472    0.472
    LS                0.405    0.057    7.132    0.000    0.650    0.650
  SUP_Friends ~~                                                        
    SE_Academic       0.071    0.047    1.493    0.136    0.100    0.100
    SE_Social         0.196    0.046    4.281    0.000    0.301    0.301
    LS                0.175    0.051    3.445    0.001    0.257    0.257
  SE_Academic ~~                                                        
    SE_Social         0.271    0.036    7.493    0.000    0.611    0.611
    LS                0.238    0.039    6.065    0.000    0.514    0.514
  SE_Social ~~                                                          
    LS                0.321    0.042    7.659    0.000    0.755    0.755

Variances:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
   .sup_parents_p1    0.273    0.037    7.358    0.000    0.273    0.238
   .sup_parents_p2    0.255    0.038    6.738    0.000    0.255    0.213
   .sup_parents_p3    0.437    0.048    9.201    0.000    0.437    0.335
   .sup_friends_p1    0.227    0.040    5.656    0.000    0.227    0.179
   .sup_friends_p2    0.238    0.030    7.936    0.000    0.238    0.266
   .sup_friends_p3    0.371    0.042    8.809    0.000    0.371    0.310
   .se_acad_p1        0.144    0.022    6.593    0.000    0.144    0.229
   .se_acad_p2        0.153    0.018    8.621    0.000    0.153    0.327
   .se_acad_p3        0.200    0.024    8.372    0.000    0.200    0.313
   .se_social_p1      0.166    0.020    8.134    0.000    0.166    0.290
   .se_social_p2      0.106    0.016    6.542    0.000    0.106    0.217
   .se_social_p3      0.288    0.028   10.132    0.000    0.288    0.451
   .ls_p1             0.417    0.045    9.233    0.000    0.417    0.484
   .ls_p2             0.261    0.028    9.321    0.000    0.261    0.492
   .ls_p3             0.362    0.040    9.005    0.000    0.362    0.465
    SUP_Parents       0.875    0.098    8.910    0.000    1.000    1.000
    SUP_Friends       1.042    0.111    9.407    0.000    1.000    1.000
    SE_Academic       0.483    0.054    8.880    0.000    1.000    1.000
    SE_Social         0.407    0.048    8.403    0.000    1.000    1.000
    LS                0.444    0.069    6.394    0.000    1.000    1.000
g1 = plot_heatmap(cov(df[,  drivers]))

g2 = plot_heatmap(data.frame(fitted(fit_mod)$cov), title="Model Implied Covariances", "Fitted Values")

plot <- ggarrange(g1,g2, ncol=1, nrow=2);

Modification Indices

modindices(fit_mod, sort = TRUE, maximum.number = 6, standardized = FALSE, minimum.value = 5)
               lhs op            rhs     mi    epc
57     SUP_Parents =~          ls_p3 28.086  0.411
152 sup_friends_p1 ~~   se_social_p3 18.792  0.094
64     SUP_Friends =~   se_social_p1 16.789 -0.134
60     SUP_Friends =~ sup_parents_p3 16.443 -0.189
68     SUP_Friends =~          ls_p2 15.119  0.155
210          ls_p2 ~~          ls_p3 13.346 -0.105

Summary Global Fit Measures

summary_df = cbind(fitMeasures(fit_mod, c("chisq", "baseline.chisq", "cfi", "aic", "bic", "rmsea","srmr")),
      fitMeasures(fit_mod_1, c("chisq", "baseline.chisq", "cfi", "aic", "bic", "rmsea","srmr")))
colnames(summary_df) = c('Full Model', 'Reduced Model')

summary_df |> kable() |> kable_styling() |>  kable_classic(full_width = F, html_font = "Cambria")
Full Model Reduced Model
chisq 223.9922306 242.1450133
baseline.chisq 2696.4887420 2696.4887420
cfi 0.9444365 0.9382035
aic 8651.9435210 8666.0963037
bic 8797.7613969 8804.6232858
rmsea 0.0797501 0.0830724
srmr 0.0558656 0.0587591
semPlot::semPaths(fit_mod, whatLabels = 'est', intercepts = FALSE)

semPlot::semPaths(fit_mod_1, whatLabels = 'std', intercepts = FALSE)

Comparing the empirical and implied variance-covariance matrix

heat_df = data.frame(resid(fit_mod, type = "standardized")$cov) 
heat_df = heat_df |> as.matrix() |> melt()
colnames(heat_df) <- c("x", "y", "value")

g1 = heat_df  |> mutate(value = round(value, 2)) |>  ggplot(aes(x = x, y = y, fill = value)) +
  geom_tile() + geom_text(aes(label = value), color = "black", size = 4) +
 scale_fill_gradient2(
    high = 'dodgerblue4',
    mid = 'white',
    low = 'firebrick2'
  ) + theme(axis.text.x = element_text(angle=45)) + ggtitle("Residuals of the Sample Covariances and Model Implied Covariances", "A Visual Check of Model fit: Full Model")

heat_df = data.frame(resid(fit_mod_1, type = "standardized")$cov) 
heat_df = heat_df |> as.matrix() |> melt()
colnames(heat_df) <- c("x", "y", "value")

g2 = heat_df  |> mutate(value = round(value, 2)) |>  ggplot(aes(x = x, y = y, fill = value)) +
  geom_tile() + geom_text(aes(label = value), color = "black", size = 4) +
 scale_fill_gradient2(
    high = 'dodgerblue4',
    mid = 'white',
    low = 'firebrick2'
  ) + theme(axis.text.x = element_text(angle=45)) + ggtitle("Residuals of the Sample Covariances and Model Implied Covariances", "A Visual Check of Model fit: Reduced Model")

plot <- ggarrange(g1,g2, ncol=1, nrow=2);

anova(fit_mod)
Chi-Squared Test Statistic (unscaled)

          Df    AIC    BIC  Chisq Chisq diff Df diff           Pr(>Chisq)    
Saturated  0                 0.00                                            
Model     80 8651.9 8797.8 223.99     223.99      80 0.000000000000001443 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
anova(fit_mod_1)
Chi-Squared Test Statistic (unscaled)

          Df    AIC    BIC  Chisq Chisq diff Df diff            Pr(>Chisq)    
Saturated  0                 0.00                                             
Model     82 8666.1 8804.6 242.15     242.15      82 < 0.00000000000000022 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
anova(fit_mod, fit_mod_1)

Chi-Squared Difference Test

          Df    AIC    BIC  Chisq Chisq diff   RMSEA Df diff Pr(>Chisq)    
fit_mod   80 8651.9 8797.8 223.99                                          
fit_mod_1 82 8666.1 8804.6 242.15     18.153 0.16893       2  0.0001143 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Structural Equation Models

model <- "
# Measurement model
SUP_Parents =~ sup_parents_p1 + sup_parents_p2 + sup_parents_p3
SUP_Friends =~ sup_friends_p1 + sup_friends_p2 + sup_friends_p3
SE_Academic =~ se_acad_p1 + se_acad_p2 + se_acad_p3
SE_Social =~ se_social_p1 + se_social_p2 + se_social_p3
LS  =~ ls_p1 + ls_p2 + ls_p3

# Structural model 
# Regressions
LS ~ SE_Academic + SE_Social + SUP_Parents + SUP_Friends

# Residual covariances
SE_Academic ~~ SE_Social
"

fit_mod_sem <- sem(model, data = df)
modelplot(fit_mod_sem)

semPlot::semPaths(fit_mod_sem, whatLabels = 'std', intercepts = FALSE)

heat_df = data.frame(resid(fit_mod_sem, type = "standardized")$cov) 
heat_df = heat_df |> as.matrix() |> melt()
colnames(heat_df) <- c("x", "y", "value")

heat_df  |> mutate(value = round(value, 2)) |>  ggplot(aes(x = x, y = y, fill = value)) +
  geom_tile() + geom_text(aes(label = value), color = "black", size = 4) +
 scale_fill_gradient2(
    high = 'dodgerblue4',
    mid = 'white',
    low = 'firebrick2'
  ) + theme(axis.text.x = element_text(angle=45)) + ggtitle("Residuals of the Sample Covariances and Model Implied Covariances", "A Visual Check of Model fit")

Confirmatory Factor Models with PyMC

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from pytensor import tensor as pt
import arviz as az
import networkx as nx
np.random.seed(150)



df_p = pd.read_csv('IIS.dat', sep='\s+')
df_p.head() 
     PI    AD   IGC   FI   FC
0  4.00  3.38  4.67  2.6  3.2
1  2.57  3.00  3.50  2.4  2.8
2  2.29  3.29  4.83  2.0  3.4
3  2.43  3.63  4.33  3.6  3.8
4  3.00  4.00  4.83  3.4  3.8
coords = {'obs': list(range(len(df_p))), 
          'indicators': ['PI', 'AD',    'IGC', 'FI', 'FC'],
          'indicators_1': ['PI', 'AD',  'IGC'],
          'indicators_2': ['FI', 'FC'],
          'latent': ['Student', 'Faculty']
          }


obs_idx = list(range(len(df_p)))
with pm.Model(coords=coords) as model:
  
  Psi = pm.InverseGamma('Psi', 5, 10, dims='indicators')
  lambdas_ = pm.Normal('lambdas_1', 1, 10, dims=('indicators_1'))
  lambdas_1 = pm.Deterministic('lambdas1', pt.set_subtensor(lambdas_[0], 1), dims=('indicators_1'))
  lambdas_ = pm.Normal('lambdas_2', 1, 10, dims=('indicators_2'))
  lambdas_2 = pm.Deterministic('lambdas2', pt.set_subtensor(lambdas_[0], 1), dims=('indicators_2'))
  tau = pm.Normal('tau', 3, 10, dims='indicators')
  kappa = 0
  sd_dist = pm.Exponential.dist(1.0, shape=2)
  chol, _, _ = pm.LKJCholeskyCov('chol_cov', n=2, eta=2,
    sd_dist=sd_dist, compute_corr=True)
  ksi = pm.MvNormal('ksi', kappa, chol=chol, dims=('obs', 'latent'))

  m1 = tau[0] + ksi[obs_idx, 0]*lambdas_1[0]
  m2 = tau[1] + ksi[obs_idx, 0]*lambdas_1[1]
  m3 = tau[2] + ksi[obs_idx, 0]*lambdas_1[2]
  m4 = tau[3] + ksi[obs_idx, 1]*lambdas_2[0]
  m5 = tau[4] + ksi[obs_idx, 1]*lambdas_2[1]
  
  mu = pm.Deterministic('mu', pm.math.stack([m1, m2, m3, m4, m5]).T)
  _  = pm.Normal('likelihood', mu, Psi, observed=df_p.values)

  idata = pm.sample(nuts_sampler='numpyro', target_accept=.95, 
                    idata_kwargs={"log_likelihood": True})
  idata.extend(pm.sample_posterior_predictive(idata))
  
summary_df = az.summary(idata, var_names=['lambdas1', 'lambdas2', 'tau', 'Psi', 'ksi', 'chol_cov_corr'], coords= {'obs': [0, 7]})

PyMC Confirmatory Factor Model

py$summary_df |> kable() |>  kable_classic(full_width = F, html_font = "Cambria")
mean sd hdi_3% hdi_97% mcse_mean mcse_sd ess_bulk ess_tail r_hat
lambdas1[PI] 1.000 0.000 1.000 1.000 0.000 0.000 4000 4000 NaN
lambdas1[AD] 0.898 0.062 0.784 1.015 0.004 0.003 267 524 1.02
lambdas1[IGC] 0.536 0.045 0.452 0.619 0.002 0.002 394 799 1.02
lambdas2[FI] 1.000 0.000 1.000 1.000 0.000 0.000 4000 4000 NaN
lambdas2[FC] 0.980 0.058 0.876 1.091 0.003 0.002 443 943 1.00
tau[PI] 3.332 0.037 3.262 3.398 0.002 0.001 563 1555 1.00
tau[AD] 3.897 0.026 3.850 3.948 0.001 0.001 387 1008 1.00
tau[IGC] 4.596 0.020 4.558 4.634 0.001 0.001 666 1582 1.00
tau[FI] 3.033 0.039 2.961 3.108 0.002 0.001 476 1158 1.00
tau[FC] 3.712 0.034 3.650 3.779 0.002 0.001 374 922 1.00
Psi[PI] 0.610 0.024 0.565 0.654 0.001 0.000 1250 2084 1.00
Psi[AD] 0.318 0.020 0.282 0.355 0.001 0.001 484 1118 1.00
Psi[IGC] 0.355 0.013 0.330 0.380 0.000 0.000 2184 2208 1.00
Psi[FI] 0.570 0.026 0.522 0.620 0.001 0.001 1008 1854 1.00
Psi[FC] 0.419 0.026 0.374 0.471 0.001 0.001 694 1344 1.00
ksi[0, Student] -0.225 0.224 -0.633 0.197 0.003 0.003 4431 3395 1.00
ksi[0, Faculty] -0.370 0.277 -0.918 0.134 0.004 0.003 4265 2674 1.00
ksi[7, Student] 0.881 0.232 0.430 1.302 0.004 0.003 3152 2768 1.00
ksi[7, Faculty] 0.863 0.275 0.318 1.347 0.004 0.003 3963 2620 1.00
chol_cov_corr[0, 0] 1.000 0.000 1.000 1.000 0.000 0.000 4000 4000 NaN
chol_cov_corr[0, 1] 0.850 0.028 0.797 0.905 0.001 0.001 489 580 1.01
chol_cov_corr[1, 0] 0.850 0.028 0.797 0.905 0.001 0.001 489 580 1.01
chol_cov_corr[1, 1] 1.000 0.000 1.000 1.000 0.000 0.000 3910 3911 1.00
az.plot_trace(idata, var_names=['lambdas1', 'lambdas2', 'tau', 'Psi', 'ksi']);
/Users/nathanielforde/mambaforge/envs/pymc_causal/lib/python3.11/site-packages/arviz/stats/density_utils.py:487: UserWarning: Your data appears to have a single value or no finite values
  warnings.warn("Your data appears to have a single value or no finite values")
/Users/nathanielforde/mambaforge/envs/pymc_causal/lib/python3.11/site-packages/arviz/stats/density_utils.py:487: UserWarning: Your data appears to have a single value or no finite values
  warnings.warn("Your data appears to have a single value or no finite values")


df = pd.read_csv('sem_data.csv')
drivers = ['se_acad_p1', 'se_acad_p2',
       'se_acad_p3', 'se_social_p1', 'se_social_p2', 'se_social_p3',
       'sup_friends_p1', 'sup_friends_p2', 'sup_friends_p3', 'sup_parents_p1',
       'sup_parents_p2', 'sup_parents_p3', 'ls_p1', 'ls_p2', 'ls_p3']
       
coords = {'obs': list(range(len(df))), 
          'indicators': drivers,
          'indicators_1': ['se_acad_p1','se_acad_p2','se_acad_p3'],
          'indicators_2': ['se_social_p1','se_social_p2','se_social_p3'],
          'indicators_3': ['sup_friends_p1','sup_friends_p2','sup_friends_p3'],
          'indicators_4': [ 'sup_parents_p1','sup_parents_p2','sup_parents_p3'],
          'indicators_5': ['ls_p1','ls_p2', 'ls_p3'],
          'latent': ['SE_ACAD', 'SE_SOCIAL', 'SUP_F', 'SUP_P', 'LS'],
          'latent1': ['SE_ACAD', 'SE_SOCIAL', 'SUP_F', 'SUP_P', 'LS']
          }

obs_idx = list(range(len(df)))
with pm.Model(coords=coords) as model:
  
  Psi = pm.InverseGamma('Psi', 5, 10, dims='indicators')
  lambdas_ = pm.Normal('lambdas_1', 1, 10, dims=('indicators_1'))
  lambdas_1 = pm.Deterministic('lambdas1', pt.set_subtensor(lambdas_[0], 1), dims=('indicators_1'))
  lambdas_ = pm.Normal('lambdas_2', 1, 10, dims=('indicators_2'))
  lambdas_2 = pm.Deterministic('lambdas2', pt.set_subtensor(lambdas_[0], 1), dims=('indicators_2'))
  lambdas_ = pm.Normal('lambdas_3', 1, 10, dims=('indicators_3'))
  lambdas_3 = pm.Deterministic('lambdas3', pt.set_subtensor(lambdas_[0], 1), dims=('indicators_3'))
  lambdas_ = pm.Normal('lambdas_4', 1, 10, dims=('indicators_4'))
  lambdas_4 = pm.Deterministic('lambdas4', pt.set_subtensor(lambdas_[0], 1), dims=('indicators_4'))
  lambdas_ = pm.Normal('lambdas_5', 1, 10, dims=('indicators_5'))
  lambdas_5 = pm.Deterministic('lambdas5', pt.set_subtensor(lambdas_[0], 1), dims=('indicators_5'))
  tau = pm.Normal('tau', 3, 10, dims='indicators')
  kappa = 0
  sd_dist = pm.Exponential.dist(1.0, shape=5)
  chol, _, _ = pm.LKJCholeskyCov('chol_cov', n=5, eta=2,
    sd_dist=sd_dist, compute_corr=True)
  cov = pm.Deterministic("cov", chol.dot(chol.T), dims=('latent', 'latent1'))
  ksi = pm.MvNormal('ksi', kappa, chol=chol, dims=('obs', 'latent'))

  m0 = tau[0] + ksi[obs_idx, 0]*lambdas_1[0]
  m1 = tau[1] + ksi[obs_idx, 0]*lambdas_1[1]
  m2 = tau[2] + ksi[obs_idx, 0]*lambdas_1[2]
  m3 = tau[3] + ksi[obs_idx, 1]*lambdas_2[0]
  m4 = tau[4] + ksi[obs_idx, 1]*lambdas_2[1]
  m5 = tau[5] + ksi[obs_idx, 1]*lambdas_2[2]
  m6 = tau[6] + ksi[obs_idx, 2]*lambdas_3[0]
  m7 = tau[7] + ksi[obs_idx, 2]*lambdas_3[1]
  m8 = tau[8] + ksi[obs_idx, 2]*lambdas_3[2]
  m9 = tau[9] + ksi[obs_idx, 3]*lambdas_4[0]
  m10 = tau[10] + ksi[obs_idx, 3]*lambdas_4[1]
  m11 = tau[11] + ksi[obs_idx, 3]*lambdas_4[2]
  m12 = tau[12] + ksi[obs_idx, 4]*lambdas_5[0]
  m13 = tau[13] + ksi[obs_idx, 4]*lambdas_5[1]
  m14 = tau[14] + ksi[obs_idx, 4]*lambdas_5[2]
  
  mu = pm.Deterministic('mu', pm.math.stack([m0, m1, m2, m3, m4, m5, m6, m7,
                                             m8, m9, m10, m11, m12, m13, m14]).T)
  _  = pm.Normal('likelihood', mu, Psi, observed=df[drivers].values)

  idata = pm.sample(nuts_sampler='numpyro', target_accept=.95, tune=1000,
                    idata_kwargs={"log_likelihood": True}, random_seed=100)
  idata.extend(pm.sample_posterior_predictive(idata))
  
summary_df1 = az.summary(idata, var_names=['lambdas1', 'lambdas2', 'lambdas3', 'lambdas4', 'lambdas5', 'tau', 'Psi'])

cov_df = pd.DataFrame(az.extract(idata['posterior'])['cov'].mean(axis=2))
cov_df.index = ['SE_ACAD', 'SE_SOCIAL', 'SUP_F', 'SUP_P', 'LS']
cov_df.columns = ['SE_ACAD', 'SE_SOCIAL', 'SUP_F', 'SUP_P', 'LS']

correlation_df = pd.DataFrame(az.extract(idata['posterior'])['chol_cov_corr'].mean(axis=2))
correlation_df.index = ['SE_ACAD', 'SE_SOCIAL', 'SUP_F', 'SUP_P', 'LS']
correlation_df.columns = ['SE_ACAD', 'SE_SOCIAL', 'SUP_F', 'SUP_P', 'LS']

Life Satisfaction Model

py$summary_df1 |> kable() |>  kable_classic(full_width = F, html_font = "Cambria")
mean sd hdi_3% hdi_97% mcse_mean mcse_sd ess_bulk ess_tail r_hat
lambdas1[se_acad_p1] 1.000 0.000 1.000 1.000 0.000 0.000 4000 4000 NaN
lambdas1[se_acad_p2] 0.817 0.052 0.720 0.915 0.001 0.001 1193 2008 1.00
lambdas1[se_acad_p3] 0.967 0.060 0.854 1.076 0.002 0.001 1286 2037 1.00
lambdas2[se_social_p1] 1.000 0.000 1.000 1.000 0.000 0.000 4000 4000 NaN
lambdas2[se_social_p2] 0.965 0.058 0.856 1.071 0.002 0.002 757 1634 1.00
lambdas2[se_social_p3] 0.941 0.072 0.805 1.076 0.002 0.002 878 1580 1.00
lambdas3[sup_friends_p1] 1.000 0.000 1.000 1.000 0.000 0.000 4000 4000 NaN
lambdas3[sup_friends_p2] 0.802 0.044 0.720 0.887 0.001 0.001 1045 1701 1.00
lambdas3[sup_friends_p3] 0.905 0.053 0.805 1.006 0.002 0.001 1235 2150 1.00
lambdas4[sup_parents_p1] 1.000 0.000 1.000 1.000 0.000 0.000 4000 4000 NaN
lambdas4[sup_parents_p2] 1.040 0.059 0.931 1.150 0.002 0.002 758 1383 1.00
lambdas4[sup_parents_p3] 1.010 0.064 0.898 1.137 0.002 0.001 1051 1840 1.00
lambdas5[ls_p1] 1.000 0.000 1.000 1.000 0.000 0.000 4000 4000 NaN
lambdas5[ls_p2] 0.791 0.085 0.627 0.944 0.004 0.003 541 1074 1.00
lambdas5[ls_p3] 0.990 0.103 0.806 1.187 0.004 0.003 543 878 1.00
tau[se_acad_p1] 5.153 0.044 5.069 5.234 0.002 0.001 488 1151 1.01
tau[se_acad_p2] 5.345 0.039 5.271 5.414 0.002 0.001 528 1033 1.01
tau[se_acad_p3] 5.209 0.045 5.127 5.297 0.002 0.001 526 1290 1.01
tau[se_social_p1] 5.286 0.042 5.208 5.366 0.002 0.002 380 743 1.01
tau[se_social_p2] 5.473 0.039 5.397 5.544 0.002 0.001 363 742 1.01
tau[se_social_p3] 5.437 0.045 5.351 5.522 0.002 0.001 492 982 1.00
tau[sup_friends_p1] 5.782 0.068 5.651 5.904 0.004 0.003 333 763 1.01
tau[sup_friends_p2] 6.007 0.057 5.909 6.125 0.003 0.002 397 872 1.00
tau[sup_friends_p3] 5.987 0.066 5.864 6.115 0.003 0.002 385 890 1.01
tau[sup_parents_p1] 5.973 0.061 5.858 6.085 0.003 0.002 427 1059 1.00
tau[sup_parents_p2] 5.925 0.062 5.807 6.040 0.003 0.002 394 924 1.01
tau[sup_parents_p3] 5.716 0.066 5.596 5.840 0.003 0.002 470 1294 1.00
tau[ls_p1] 5.188 0.053 5.092 5.289 0.002 0.001 654 1378 1.00
tau[ls_p2] 5.775 0.041 5.693 5.849 0.002 0.001 716 1596 1.00
tau[ls_p3] 5.219 0.051 5.121 5.314 0.002 0.001 666 1609 1.00
Psi[se_acad_p1] 0.412 0.028 0.359 0.465 0.001 0.001 1278 1740 1.00
Psi[se_acad_p2] 0.413 0.024 0.367 0.456 0.001 0.000 2170 2268 1.00
Psi[se_acad_p3] 0.468 0.027 0.418 0.519 0.001 0.000 1844 2408 1.00
Psi[se_social_p1] 0.431 0.026 0.381 0.477 0.001 0.000 1382 2219 1.00
Psi[se_social_p2] 0.361 0.025 0.314 0.405 0.001 0.000 1486 2135 1.00
Psi[se_social_p3] 0.553 0.029 0.500 0.606 0.001 0.000 2594 2803 1.00
Psi[sup_friends_p1] 0.517 0.040 0.439 0.587 0.001 0.001 866 1739 1.00
Psi[sup_friends_p2] 0.508 0.031 0.454 0.568 0.001 0.001 1420 1985 1.00
Psi[sup_friends_p3] 0.625 0.036 0.562 0.694 0.001 0.001 2090 2329 1.00
Psi[sup_parents_p1] 0.550 0.035 0.485 0.615 0.001 0.001 1530 2075 1.00
Psi[sup_parents_p2] 0.536 0.038 0.465 0.605 0.001 0.001 1192 2078 1.01
Psi[sup_parents_p3] 0.675 0.038 0.602 0.745 0.001 0.001 2089 2371 1.00
Psi[ls_p1] 0.671 0.038 0.603 0.744 0.001 0.001 1045 2387 1.00
Psi[ls_p2] 0.534 0.030 0.477 0.591 0.001 0.001 1409 2472 1.00
Psi[ls_p3] 0.622 0.035 0.554 0.688 0.001 0.001 1597 2393 1.00

fig, ax = plt.subplots(figsize=(15, 8))
az.plot_forest(idata, var_names=['lambdas1', 'lambdas2', 'lambdas3', 'lambdas4', 'lambdas5'], combined=True, ax=ax);
ax.set_title("Factor Loadings for each of the Five Factors");

py$cov_df |> kable(caption= "Covariances Amongst Latent Factors",digits=2) |> kable_styling() %>% kable_classic(full_width = F, html_font = "Cambria") |> row_spec(5, color = "red")
Covariances Amongst Latent Factors
SE_ACAD SE_SOCIAL SUP_F SUP_P LS
SE_ACAD 0.47 0.26 0.06 0.20 0.22
SE_SOCIAL 0.26 0.39 0.19 0.26 0.30
SUP_F 0.06 0.19 1.03 0.12 0.16
SUP_P 0.20 0.26 0.12 0.86 0.38
LS 0.22 0.30 0.16 0.38 0.42
py$correlation_df |> kable( caption= "Correlations Amongst Latent Factors", digits=2) |> kable_styling() |>  kable_classic(full_width = F, html_font = "Cambria") |> row_spec(5, color = "red")
Correlations Amongst Latent Factors
SE_ACAD SE_SOCIAL SUP_F SUP_P LS
SE_ACAD 1.00 0.60 0.09 0.32 0.50
SE_SOCIAL 0.60 1.00 0.29 0.45 0.75
SUP_F 0.09 0.29 1.00 0.12 0.25
SUP_P 0.32 0.45 0.12 1.00 0.64
LS 0.50 0.75 0.25 0.64 1.00
def make_ppc(idata):
  fig, axs = plt.subplots(5, 3, figsize=(20, 20))
  axs = axs.flatten()
  for i in range(15):
      temp = idata['posterior_predictive'].sel({'likelihood_dim_3': i}).mean(dim=('chain', 'draw'))
      axs[i].hist(df[drivers[i]], alpha=0.3, ec='black', bins=20, label='Observed Scores')
      axs[i].hist(temp['likelihood'], color='purple', alpha=0.6, bins=20, label='Predicted Scores')
      axs[i].set_title(f"Posterior Predictive Checks {drivers[i]}")
      axs[i].legend();
  plt.show()
  
make_ppc(idata)

model_cov = pd.DataFrame(az.extract(idata['posterior_predictive'])['likelihood'][:, :, 0]).cov()
obs_cov = df[drivers].cov()
model_cov.index = obs_cov.index
model_cov.columns = obs_cov.columns
residuals = model_cov - obs_cov
plot_heatmap(py$residuals, "Residuals of the Sample Covariances and Model Implied Covariances", "A Visual Check of Bayesian Model fit")

Structural Equation Modelling in PyMC

def make_sem(priors): 

  coords = {'obs': list(range(len(df))), 
            'indicators': drivers,
            'indicators_1': ['se_acad_p1','se_acad_p2','se_acad_p3'],
            'indicators_2': ['se_social_p1','se_social_p2','se_social_p3'],
            'indicators_3': ['sup_friends_p1','sup_friends_p2','sup_friends_p3'],
            'indicators_4': [ 'sup_parents_p1','sup_parents_p2','sup_parents_p3'],
            'indicators_5': ['ls_p1','ls_p2', 'ls_p3'],
            'latent': ['SE_ACAD', 'SE_SOCIAL', 'SUP_F', 'SUP_P'], 
            'latent1': ['SE_ACAD', 'SE_SOCIAL', 'SUP_F', 'SUP_P']
            }

  obs_idx = list(range(len(df)))
  with pm.Model(coords=coords) as model:
    
    Psi = pm.InverseGamma('Psi', 5, 10, dims='indicators')
    lambdas_ = pm.Normal('lambdas_1',  priors['lambda'][0], priors['lambda'][1], dims=('indicators_1'))
    lambdas_1 = pm.Deterministic('lambdas1', pt.set_subtensor(lambdas_[0], 1), dims=('indicators_1'))
    lambdas_ = pm.Normal('lambdas_2', priors['lambda'][0], priors['lambda'][1], dims=('indicators_2'))
    lambdas_2 = pm.Deterministic('lambdas2', pt.set_subtensor(lambdas_[0], 1), dims=('indicators_2'))
    lambdas_ = pm.Normal('lambdas_3', priors['lambda'][0], priors['lambda'][1], dims=('indicators_3'))
    lambdas_3 = pm.Deterministic('lambdas3', pt.set_subtensor(lambdas_[0], 1), dims=('indicators_3'))
    lambdas_ = pm.Normal('lambdas_4', priors['lambda'][0], priors['lambda'][1], dims=('indicators_4'))
    lambdas_4 = pm.Deterministic('lambdas4', pt.set_subtensor(lambdas_[0], 1), dims=('indicators_4'))
    lambdas_ = pm.Normal('lambdas_5', priors['lambda'][0], priors['lambda'][1], dims=('indicators_5'))
    lambdas_5 = pm.Deterministic('lambdas5', pt.set_subtensor(lambdas_[0], 1), dims=('indicators_5'))
    tau = pm.Normal('tau', 3, 10, dims='indicators')
    kappa = 0
    sd_dist = pm.Exponential.dist(1.0, shape=4)
    chol, _, _ = pm.LKJCholeskyCov('chol_cov', n=4, eta=priors['eta'],
      sd_dist=sd_dist, compute_corr=True)
    cov = pm.Deterministic("cov", chol.dot(chol.T), dims=('latent', 'latent1'))
    ksi = pm.MvNormal('ksi', kappa, chol=chol, dims=('obs', 'latent'))

    beta_r = pm.Normal('beta_r', 0, 1, dims='latent')
    regression = pm.Deterministic('regr', beta_r[0]*ksi[obs_idx, 0] + beta_r[1]*ksi[obs_idx, 1] +
                                   beta_r[2]*ksi[obs_idx, 2] + beta_r[3]*ksi[obs_idx, 3])

    m0 = tau[0] + ksi[obs_idx, 0]*lambdas_1[0]
    m1 = tau[1] + ksi[obs_idx, 0]*lambdas_1[1]
    m2 = tau[2] + ksi[obs_idx, 0]*lambdas_1[2]
    m3 = tau[3] + ksi[obs_idx, 1]*lambdas_2[0]
    m4 = tau[4] + ksi[obs_idx, 1]*lambdas_2[1]
    m5 = tau[5] + ksi[obs_idx, 1]*lambdas_2[2]
    m6 = tau[6] + ksi[obs_idx, 2]*lambdas_3[0]
    m7 = tau[7] + ksi[obs_idx, 2]*lambdas_3[1]
    m8 = tau[8] + ksi[obs_idx, 2]*lambdas_3[2]
    m9 = tau[9] + ksi[obs_idx, 3]*lambdas_4[0]
    m10 = tau[10] + ksi[obs_idx, 3]*lambdas_4[1]
    m11 = tau[11] + ksi[obs_idx, 3]*lambdas_4[2]
    m12 = tau[12] + regression*lambdas_5[0]
    m13 = tau[13] + regression*lambdas_5[1]
    m14 = tau[14] + regression*lambdas_5[2]
    
    mu = pm.Deterministic('mu', pm.math.stack([m0, m1, m2, m3, m4, m5, m6, m7,
                                              m8, m9, m10, m11, m12, m13, m14]).T)
    _  = pm.Normal('likelihood', mu, Psi, observed=df[drivers].values)

    idata = pm.sample(chains=6, nuts_sampler='numpyro', target_accept=.95, tune=1000,
                      idata_kwargs={"log_likelihood": True}, random_seed=100)
    idata.extend(pm.sample_posterior_predictive(idata))

    return model, idata


model1, idata1 = make_sem(priors={'eta': 2, 'lambda': [1, 2]})

Structural Equation Models

az.plot_posterior(idata1, var_names=['beta_r']);

make_ppc(idata1)

Citation

BibTeX citation:
@online{forde,
  author = {Nathaniel Forde},
  title = {Measurement, {Latent} {Factors} and {Theory} {Construction}},
  langid = {en}
}
For attribution, please cite this work as:
Nathaniel Forde. n.d. “Measurement, Latent Factors and Theory Construction.”